The battle over those differences now moves to the House, where Republicans are pushing an alternative version of the bill.

The Senate voted 68 to 31 to pass the bill following an attempt by Democrats in recent weeks to paint Republican objections as a new assault on issues important to women. By Thursday, Republicans were insisting they also wished for speedy passage of the bill, despite their concerns.

Republican leaders had urged the Democrats who control floor action to allow votes on a GOP-authored alternative to the bill, as well as an amendment by Sen. John Cornyn (R-Tex.) that would earmark new funds for testing a backlog of rape kits and impose new mandatory minimum sentences for perpetrators of some domestic violence crimes, including possession of child pornography. Both GOP measures failed Thursday.

The Senate bill was crafted in consultation with law enforcement groups and victims advocates and was introduced with a filibuster-proof 61 co-sponsors, including eight Republicans.

Democrats believed that gave Republicans who oppose the measure little leverage to demand votes on amendments likely to fail. They objected to Cornyn’s proposal because some advocates believe requiring judges to impose harsh mandatory sentences on abusers can sometimes discourage victims from reporting their crimes.

First enacted in 1994, the Violence Against Women Act was reauthorized in 2000 and again in 2005. The measure funds grant programs that help local law enforcement in identifying and prosecuting domestic abuse and sexual violence, as well as programs that assist victims.

The new bill would restructure and consolidate grant programs and create new programs to raise awareness on college campuses.

“I do believe it represents a real improvement in the services offered to victims, even in a difficult budget environment,” she said.

But other Republicans objected to a number of the measure’s new provisions. One would add language barring discrimination on the basis of sexual orientation and gender identity in programs funded through the measure.

Another would let law enforcement issue up to 5,000 more visas each year to battered illegal immigrants who agree to participate in the prosecution of serious crimes. The 2000 update of the bill set aside 10,000 visas annually for that purpose, which advocates believe encourages victims to report crime. All 10,000 are being issued each year, and advocates say more are needed.

A final area of contention would provide the government new authority to prosecute non-Indian men who abuse Indian women on tribal reservations.

The White House called on Congress to pass the measure Thursday. Senior Adviser Valerie Jarrett said the bill had “transformed the nation’s response to violence against women” and said she was concerned Republican alternatives would leave “too many victims without protections.”

In the House, a group of Republican women announced Wednesday that they will introduce an alternative version of the legislation. The differences between the competing bills will have to be worked out in coming months. Those negotiations will probably feature attempts by both parties to seek political advantage while accusing the other of politicizing the sensitive issue of domestic and sexual violence.